##
**Algebraic function fields and codes.**
*(English)*
Zbl 0816.14011

Universitext. Berlin: Springer-Verlag. x, 260 p. (1993).

This book is an introduction to the theory of algebraic curves from an algebraic point of view with applications to coding theory. Since a nonsingular projective curve over a field is completely determined by its function field it is possible to study algebraic curves in a purely algebraic way through valuations and fields. In this sense the book fits in a German tradition which starts with Weber and runs via Kronecker to Hasse and Deuring. The disadvantage of this one-sidedness is that no methods which are based on geometry in higher dimensions, such as the theory of jacobians, come on the screen.

The author gives a conscientious description of the theory of algebraic curves: the Riemann-Roch theorem, extensions of function fields, differentials, curves over finite fields, and the Hasse-Weil theorem are treated. As far as applications to coding theory are concerned attention is paid to geometric Goppa codes and trace codes. Geometric Goppa codes are codes which are defined by evaluating functions from a linear system on a fixed curve over a finite field, in rational points on the curve. Properties of the parameters of these codes follow from properties of the linear systems involved. Trace codes over a finite field \(\mathbb{F}_ q\) are obtained by applying the trace map from \(\mathbb{F}_{q^ m}\) to \(\mathbb{F}_ q\) to a linear code over \(\mathbb{F}_{q^ m}\). Classical codes, such as cyclic codes, can be viewed as trace codes. The nonzero words in a trace code correspond to algebraic curves, in fact Artin-Schreier curves, and results on these curves lead to results on trace codes.

The book is very carefully written and amply provided with illustrative examples. Especially those who prefer the purely algebraic approach to the theory of algebraic curves will appreciate the book.

The author gives a conscientious description of the theory of algebraic curves: the Riemann-Roch theorem, extensions of function fields, differentials, curves over finite fields, and the Hasse-Weil theorem are treated. As far as applications to coding theory are concerned attention is paid to geometric Goppa codes and trace codes. Geometric Goppa codes are codes which are defined by evaluating functions from a linear system on a fixed curve over a finite field, in rational points on the curve. Properties of the parameters of these codes follow from properties of the linear systems involved. Trace codes over a finite field \(\mathbb{F}_ q\) are obtained by applying the trace map from \(\mathbb{F}_{q^ m}\) to \(\mathbb{F}_ q\) to a linear code over \(\mathbb{F}_{q^ m}\). Classical codes, such as cyclic codes, can be viewed as trace codes. The nonzero words in a trace code correspond to algebraic curves, in fact Artin-Schreier curves, and results on these curves lead to results on trace codes.

The book is very carefully written and amply provided with illustrative examples. Especially those who prefer the purely algebraic approach to the theory of algebraic curves will appreciate the book.

Reviewer: M.van der Vlugt (Leiden)

### MSC:

14H05 | Algebraic functions and function fields in algebraic geometry |

11R58 | Arithmetic theory of algebraic function fields |

14-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to algebraic geometry |

14-02 | Research exposition (monographs, survey articles) pertaining to algebraic geometry |

94-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to information and communication theory |

94-02 | Research exposition (monographs, survey articles) pertaining to information and communication theory |

94B27 | Geometric methods (including applications of algebraic geometry) applied to coding theory |

94B15 | Cyclic codes |